# Science:Math Exam Resources/Courses/MATH257/December 2011/Question 04 (a)

MATH257 December 2011
Other MATH257 Exams

### Question 04 (a)

Consider the following problem for the heat equation with a time-dependent source term, and mixed boundary conditions:

${\displaystyle u_{t}=u_{xx}+t,\quad 00,}$

${\displaystyle u_{x}(0,t)=0,\quad u(1,t)=0,\quad u(x,0)=1.}$

Briefly describe how you would use the method of finite differences to find an approximate solution to this problem. Use the notation ${\displaystyle u_{n}^{k}\approx u(x_{n},t_{k})}$ to denote the values of ${\displaystyle u}$ on the finite difference mesh, and include how you propose to incorporate the boundary and initial conditions. In case it is useful, the Taylor expansion formula is ${\displaystyle f(x+\Delta x)=f(x)+f'(x)\Delta x+{\frac {1}{2}}f''(x)(\Delta x)^{2}+O((\Delta x)^{3}).}$

 Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

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